MA101.19 The Chain Rule – One Variable

Consider the function f:\mathbb{R} \to \mathbb{R} given by f(x) = \sin{x}. We are used to wring such things as:

i) f'
ii) f'(x)
iii) \displaystyle \frac{df}{dx}.  For example we would write f'(x) = \cos{x} for example.

Equally well, of course, it would be true to write f'(u) = \cos{u}.

The meaning of (i) and (ii) are mathematically precise. f' means the derived function and f'(x) means the value of f' at x.

The meaning of \displaystyle \frac{df}{dx} can be more devious.

It can simply be taken as synonymous with f'(x). That is \displaystyle \frac{df}{dx} = f'(x).

When such is the intention it would be indisputable that \displaystyle \frac{df}{du} means f'(u).

But there are other more shady uses as we will see.

Now consider substituting x = u^2 in f(x) = \sin{x} to define a function F defined as f(u) = \sin{u^2}.

The chain rule says

\begin{aligned} \displaystyle \frac{dF}{du} &= \frac{df}{dx} \cdot \frac{dx}{du} \\&= \cos{x} \cdot 2u \\&= 2u\cos{u^2} \end{aligned}

where here \frac{dF}{du} means F'(u).

Note that F is not equal to f, but mathematicians frequently write the chain rule as,

\displaystyle \frac{df}{du} = \frac{df}{dx} \cdot \frac{dx}{du}.

Here \frac{df}{du} does not mean f'(u) which is after all \cos{u}.

To see the chain rule in a more precise and unambiguous form think of x=u^2 as defining a function g given by g(u)=u^2, then F = f \circ g and we see the chain rule as saying

(f \circ g)'(u) = f'(g(u)) \cdot g'(u)

Of course the u here is an entirely dummy symbol.

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